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High School Math Grades 9-12, Ages 13-18

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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k a March Highlights and 2025 AoPS Online Class Information

jlacosta 0

Mar 2, 2025

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0 replies

jlacosta

Mar 2, 2025

0 replies

Quadratic Equation Problems

Saucepan_man02 4

N 13 minutes ago by Saucepan_man02

P1) Let $\alpha < \beta < \gamma$ be the roots of $ax^3+bx^2+cx+d=0$ with $7a+3b>0, 3a+2b<0$ and $\alpha(\beta+1)+\beta(\gamma+1)+\gamma(1+\alpha)=0$ .
If [.] denotes the greatest integer function, then the number of possible values of $[|3 \alpha |] + [|6 \beta|] + [|9 \gamma |]$ is equal to

4 replies

Saucepan_man02

Mar 3, 2025

Saucepan_man02

13 minutes ago

Fun combinatorics

MTA_2024 3

N 29 minutes ago by Amkan2022

A chess master plays $no$ $less$ $than$ $1$ $game$ $per$ $day$ , but no more than $10$ $games$ $per$ $week$
Prove that, if he plays long enough, we can find a series of consecutive days in which the master played $exactly$ $23$ $games$ .

3 replies

+1 w

MTA_2024

Mar 13, 2025

Amkan2022

29 minutes ago

Number theory

MTA_2024 0

an hour ago

Find all couple $(p;q)$ of primes (greater than 5) such that : $pq \mid (5^q-3^q)(5^p-3^p)$

0 replies

MTA_2024

an hour ago

0 replies

Trash Trig Sum

P_Groudon 3

N Yesterday at 7:15 PM by P_Groudon

Find the smallest positive integer $n$ such that $\sum_{k=0}^{298}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\cos(1) - \cos(n)}{2},$ where degrees are used.

3 replies

P_Groudon

Yesterday at 5:34 PM

P_Groudon

Yesterday at 7:15 PM

Geometry problem for enthusiasts

Raul_S_Baz 1

N Yesterday at 5:45 PM by Raul_S_Baz

IMAGE

1 reply

Raul_S_Baz

Yesterday at 4:51 PM

Raul_S_Baz

Yesterday at 5:45 PM

Inequalities

sqing 3

N Yesterday at 4:24 PM by DAVROS

Let $a,b,c\ge \frac{1}{2}$ and $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le 1.$ Prove that
$a+b+c\geq 2$ Let $a,b,c\ge \frac{1}{2}$ and $\left(a+\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(a+\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le \frac{9}{2}.$ Prove that
$a^2+b^2+c^2\geq 1$ Let $a,b\ge \frac{1}{2}$ and $\left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \le \frac{20}{9}.$ Prove that
$a+b\geq 2$ Let $a,b\ge \frac{1}{2}$ and $a^2+b^2=1.$ Prove that
$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$

3 replies

sqing

Mar 15, 2025

DAVROS

Yesterday at 4:24 PM

Classic geometry problem

Raul_S_Baz 2

N Yesterday at 4:10 PM by Raul_S_Baz

IMAGE

2 replies

Raul_S_Baz

Mar 4, 2025

Raul_S_Baz

Yesterday at 4:10 PM

complement counting but less rigorous

dotscom26 1

N Yesterday at 3:47 PM by dotscom26

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C, D\}$ were there in which $A$ beat $B$ , $B$ beat $C$ , $C$ beat $D$ and $D$ beat $A?$

1 reply

dotscom26

Yesterday at 3:29 PM

dotscom26

Yesterday at 3:47 PM

Trig Multiplication

szhang7 2

N Yesterday at 3:46 PM by szhang7

Find the exact value of $(2-\sin^2(\frac{\pi}{7}))(2-\sin^2(\frac{2\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))$ .

2 replies

szhang7

Sunday at 3:50 PM

szhang7

Yesterday at 3:46 PM

Inequalites

Mario16 17

N Yesterday at 3:21 PM by sqing

If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2

17 replies

Mario16

Feb 1, 2021

sqing

Yesterday at 3:21 PM

inequalities

LuvThoConBoiRoi 8

N Yesterday at 2:18 PM by sqing

With $a,b,c$ are all real positive number that: $abc=a+b+c+2$ . Prove that:
$\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} \leq \frac{3}{4}$

8 replies

LuvThoConBoiRoi

Aug 1, 2019

sqing

Yesterday at 2:18 PM

Inequality with integers and indices

Michael Niland 4

N Yesterday at 1:44 PM by krish6_9

Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$ , and for a positive integer $n\geq3$ , prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$ .

4 replies

Michael Niland

Yesterday at 6:52 AM

krish6_9

Yesterday at 1:44 PM

Complex + Radical Evaluation

Saucepan_man02 1

N Yesterday at 1:30 PM by MathRook7817

Evaluate: (without calculators)
$(\sqrt{6 - 2 \sqrt{5}} + i \sqrt{2 \sqrt{5} + 10})^5 + (\sqrt{6 - 2 \sqrt{5}} - i \sqrt{2 \sqrt{5} + 10})^5$

1 reply

Saucepan_man02

Yesterday at 1:08 PM

MathRook7817

Yesterday at 1:30 PM

Inequalities

sqing 6

N Yesterday at 9:22 AM by sqing

Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 3$ . Prove that
$a+b+a^2+b^2 \geq 28-6\sqrt{21}$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 4$ . Prove that
$a+b+a^2+b^2 \geq 10-4\sqrt 5$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 5$ . Prove that
$a+b+a^2+b^2 \geq \frac {2(26-5\sqrt{13})}{9}$ Let $a, b\geq 0$ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1}\geq 3+ \sqrt {5}$ . Prove that
$a+b+a^2+b^2 \geq 2$

6 replies

sqing

Mar 14, 2025

sqing

Yesterday at 9:22 AM

Trig Multiplication

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#1 h Sunday at 3:50 PM

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Find the exact value of $(2-\sin^2(\frac{\pi}{7}))(2-\sin^2(\frac{2\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))$ .

This post has been edited 1 time. Last edited by szhang7, Yesterday at 3:45 PM
Reason: i miscliekd

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#2 h Yesterday at 2:24 AM • 1 Y

Y by MathRook7817

Did you mean $\frac{5\pi}7$ in the end? We use difference of squares to factor $2-\sin^2x=(\sqrt2-\sin(x+\pi))(\sqrt2-\sin(x-\pi))$ . The result covers the sines of primitive seventh roots of unity. We find that
$\cos7x+i\sin7x=(\cos x+i\sin x)^7,\sin7x=7\cos^6x\sin x-35\cos^4x\sin^3x+21\cos^2x\sin^5x-\sin^7x.$ The polynomial with the sines as roots is
$7\left(1-x^2\right)^3-35\left(1-x^2\right)^2x^2+21\left(1-x^2\right)x^4-x^6,$ which has leading term $-7-35-21-1=-64$ . Plugging in $x=\sqrt2$ and dividing $-64$ gives
$\frac{-7-35\cdot2-21\cdot4-8}{-64}=\frac{169}{64}.$ Edit: okay, the solution works with 2pi/7 too

This post has been edited 1 time. Last edited by joeym2011, 3 hours ago

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164 posts

#3 h Yesterday at 3:46 PM

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mb i menat $\frac{2\pi}{7}$ i misclicked lol

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